PRMR (Pi Recursive Model of Reality) v0.03 — Mathematical Foundation

PRMR v0.03 represents the first formal mathematical foundation of the Pi Recursive Model of Reality. Building upon the recursive-state concepts introduced in v0.01 and the causal-information framework established in v0.02, this version introduces a discrete-time computational model for recursive system evolution. The framework models systems as sequences of states that undergo cycles of fragmentation, interaction, transformation, and reorganisation while preserving underlying informational continuity. A recursive operator is defined and implemented within a simulation environment, allowing the behaviour of evolving systems to be observed across multiple recursive cycles. Results demonstrate the emergence of structured complexity, hierarchical organisation, and sustained non-linear evolution under repeated recursive application. The framework is evaluated across abstract systems, physically interpreted examples, and speculative cognitive domains to examine structural consistency across different contexts. PRMR v0.03 is not presented as a complete physical theory or unified model of nature. Instead, it serves as a mathematical and computational foundation for future development. The primary goal of this version is to establish formal definitions, simulation methodology, and a testable framework that can support later empirical, comparative, and theoretical investigations. Future work includes simulation expansion, empirical validation, comparative analysis with existing scientific frameworks, and further formalisation of recursive state dynamics.